Soft Pedal, Timbre, and Normal Modes of Vibrating Strings

The soft pedal is used to change the mood of the sound from more percussive (without the soft pedal) to more serene and gentle for grand pianos (soft pedal depressed). For uprights, it mostly makes the sound softer. For grands, it should not be used solely to reduce the intensity of the sound because it will also change the timbre. In order to play pianissimo, you will just have to learn how to play softly. Another property of the grand is that very loud sounds can be made with the soft pedal depressed. The soft pedal on most uprights has only a negligibly small effect on timbre. The upright cannot produce loud sounds with the soft pedal depressed. These changes in timbre will be explained in more detail below. One difficulty with the use of the soft pedal is that it (una corda, or more correctly due corda for the modern grand) is often not indicated, so the decision to use it is often left to the pianist.
A frequently overlooked point concerning the soft pedal is hammer voicing. If you tend to need the soft pedal to play softly, or if it is distinctly easier to play pianissimo with the grand lid closed, the hammer almost certainly needs voicing. See the subsection on "Voicing" in section 7 of Chapter Two. With properly voiced hammers, you should be able to control soft playing to any desired degree without the soft pedal. With worn, compacted, hammers, playing softly is impossible and the soft pedal has much less effect in changing the tone. In that case, the soft pedal mostly helps you to play softly and the sound will have a percussive component even with the soft pedal. Therefore, with worn hammers, you lose both the ability to play softly and the truly wonderful timbre change of the soft pedal. In most cases, the original properties of the hammer can be easily restored with simple voicing (re-shaping and needling).
The uncertainties concerning the condition of the hammer are partly responsible for why the use of the soft pedal is so controversial, since many performing pianists do use it just to play softly. As shown in the section on "Voicing", energy transfer from the hammer to the string is most efficient when the string motion is still small. A compacted hammer transfers most of its energy in this range. That is why you can find so many old large grands that feel feather light. Soft hammers on the same piano (with nothing else changed), would make the action feel much heavier. This is because, with the softer impact point of the hammer, the string is lifted far from its original position before the hammer energy starts to transfer to the string. In this position, the energy transfer is more inefficient and the pianist has to push harder to produce any sound. Clearly, the effective key weight is only partly controlled by the force required to depress the key, since it also depends on the force required to produce a given amount of sound. In other words, the piano technician must strike a compromise between voicing a hammer sufficiently soft so as to produce a pleasant tone and sufficiently hard so as to produce adequate sound. For all except the highest quality pianos, the hammer needs to be on the hard side in order to produce sufficient sound and to make the action feel nimble, which makes it difficult to play softly. This in turn can "justify" use of the soft pedal where it otherwise shouldn't be used.
In most uprights, the soft pedal causes all the hammers to move closer to the strings, thus restricting hammer motion and decreasing volume. Unlike the grands, loud sounds cannot be produced in an upright when the soft pedal is depressed. One advantage of uprights is that a partial soft pedal works. There are a few uprights in which the soft pedal works similarly to that of the grands.
In modern grands, the soft pedal causes the entire action to shift to the right by one-half string distance (the distance between strings of the same note in the 3-string section). This causes the hammer to hit only two of the three strings, causing a serendipitous transformation in the character of the sound. The horizontal motion must not move one string distance because then the strings will fall into the grooves made by adjacent strings. Since string distances cannot be controlled sufficiently accurately, this would cause some strings to fall exactly into the grooves while others will miss, creating uneven sound. Also, by hitting the less used portions of the hammer between string grooves, you get an even gentler sound. In order to understand the change in timbre with the soft pedal, we must study the acoustical mechanics of coupled vibrating strings (see Scientific American reference).
Almost all of the piano sound we hear is produced by what is called normal modes in mechanics. This is the reason why the piano sound consists mostly of the fundamental and its harmonics. Normal modes can always be decomposed into components in two orthogonal planes; say, vertical and horizontal. Furthermore, these oscillations have wavelengths that are integer fractions of the string length. Why does the string oscillate in normal modes instead of producing a whole jumble of every conceivable wavelength? At the instant that the hammer hits, it does produce a lot of these. If you place your hand on the piano, you can feel the piano "shudder" for an instant. But this is like "white noise", energy spread over a wide frequency spectrum, and the component of that energy that is within the auditory range is not sufficient to produce a significant amount of what our ears interpret as sound. What happens is that most of this energy quickly escapes out of the strings through the ends, after only a few vibrations. This happens in milliseconds, too short a time for the ear to hear anything.
The only energies trapped in the strings are those in the normal modes. Why? Because in normal modes, the ends of the strings are nodes: regions of the string that do not move. Since no transverse energy can be transmitted across a string that is motionless, only the normal modes are trapped within the string. But not quite -- the ends of the piano string are not ideal (absolutely motionless) nodes. The bridge and hitch pins are designed with just enough flexibility so that a controlled amount of energy is delivered to the soundboard. This is how the piano produces the fundamental frequency and its harmonics. Only exact harmonics are trapped because these are the only vibrations whose nodes coincide with those of the fundamental at the ends of the string. Since the hammer strikes the string in the vertical plane, all the normal modes are, initially, also in the vertical plane. An inexpensive piano is not constructed as rigidly or with as heavy material as an expensive piano and therefore has looser nodes, allowing more energy to escape. Since energy escapes quickly, a cheap piano has less sustain. A larger piano can produce more sound because the longer strings, with more tension, can store more energy and, at the same time, the more rigid nodes of the heavier, better built pianos allow less energy to escape, producing a longer sustain.
What are the normal modes of three parallel strings whose ends are coupled by placing them close together at the bridge? These strings can all move in the same direction, thus pulling the piano in that direction, or move opposite to each other, in which case the piano does not move. The opposing motions are called symmetric modes because the strings move symmetrically in opposite directions about the center of gravity of the three strings. The center of gravity is stationary during these motions. Since it requires a lot of energy to move the piano, the non-symmetric modes quickly dissipate, leaving only the symmetric modes as possible normal modes of a 3-wire system. There is only one vertical normal mode for a three string system: the center string moves in one direction while the two side strings move in the opposite direction with half the amplitude. There are two horizontal normal modes: the one in which the center string is stationary and the side strings move in opposite directions, and the one in which the center string moves in one direction while the other two move in the opposite direction at half the amplitude. For a two-string system, there is no vertical normal mode! The one in which one string moves up and the other moves down is not symmetric; it twists the piano. The only possible horizontal mode for two strings is the one in which they move in opposite directions. The lack of symmetric normal modes is one reason why the fundamentals are so weak in the two and one string sections in the bass; however, they can sustain strong harmonics. The actual motion of the strings can be any combination of these normal modes. The different admixtures of the normal modes determine the polarization of the oscillations. The polarizations change with time and this change controls the nature of the piano sound, especially things like undesirable beats.
Now we can explain what happens when the hammer hits a three string system. It initially produces mostly the vertical normal modes. Since these vertical modes couple efficiently with the soundboard (which is most flexible in that direction; i.e., it is thinnest in this direction), a loud "prompt" sound is produced. Because of the high coupling efficiency, the soundboard vibrates actively like a drum, producing a drum-like percussive sound. Now, because the piano is not symmetrical on both sides of the strings, some sideways motions are created by the vertical oscillations, which transfer energy from the vertical modes into the horizontal modes. These new modes transfer energy poorly to the soundboard, which is "thickest" in the horizontal direction and cannot vibrate horizontally. This also excites a different set of vibrational modes of the soundboard, thus changing the timbre of the sound. Therefore the horizontal modes survive much longer than the vertical modes and produce the "after sound" which has a longer sustain and a different character (Scientific American article, P. 120). Therefore, when the three strings are struck, there will be a percussive prompt sound followed by a gentler after sound.
Note that the prompt sound has two components, the initial "noise bang" associated with the white noise of the hammer strike that produces large numbers of travelling waves and anharmonic vibrations, and the following prompt sound made principally by the normal modes. Because the instantaneous volume of this impact sound can be so high, it is probably this initial sound spike that is most damaging to the ear, especially from worn hammers that release most of their energies during the initial impact. See "Voicing" in section 7 of Chapter Two for details of the interaction of worn hammers with the string. For pianos with such worn hammers, it may in fact be wise to close the lid (as the majority of their owners probably do because of the painful effect on their ears). Of course, nothing beats getting the hammers properly voiced.
The above explanations are obviously greatly oversimplified. Even the Scientific American article referenced is totally inadequate in explaining the real workings of a 3-string system. That article deals mostly with the motions of one string and discusses two string interactions for ideal, simplified cases. There is no treatment of a real 3-string system. Most discussions on string vibrations are concerned with transverse motions of the strings because those motions are the most visible and they explain the existence of the fundamental and harmonics. Although nodes do not transfer transverse motions, they do transfer tensile forces. The discussions in the "Voicing" section make it clear that tensile forces cannot be ignored since they are much larger than the transverse forces and might well dominate the acoustics of the piano. Also, the conclusions of the normal mode discussions presented above depend greatly on the coupling constant. For small coupling constants, the system becomes a superposition of coupled and uncoupled motions which allows many more modes. Thus the above discussions give only a qualitative flavor of what might be happening and give neither a quantitative, nor even a correct mechanistic, description of a real piano.
This type of understanding of piano acoustics helps us find the proper ways to use the damper pedal. If the pedal is depressed before a note is played, the initial "white noise" will excite all strings, creating a soft background roar. If you place your finger on any string, you can feel it vibrate. However, octave and harmonic strings will vibrate with higher amplitudes than the dissonant strings. This indicates that the initial "white noise" is not white but favors the normal modes. This is expected because the ends of the string are held still while the hammer strikes, thus discouraging the excitation of non-normal mode vibrations. Thus the piano not only selectively traps normal modes, but also selectively generates them. Now if the pedal is depressed after the note is struck, there will be sympathetic vibration in octave and harmonic strings, but the unrelated strings will be almost totally quiet. This produces a clear sustained note. The lesson here is that, in general, the pedal should be depressed immediately after striking the note, not before. This is a good habit to cultivate.
Many of the above explanations can be proven experimentally. The motions of the strings can be measured directly by a number of readily available instruments. A second method is to make use of the fact that the string vibrations are linear processes; i.e. they decay exponentially with time. Thus when the sound decay is plotted on a logarithmic scale, you get a straight line (see Scientific American reference). However, when so plotted, one gets two straight lines, an initial line with a steep slope (faster decay), followed by another one with a less steep slope. These two lines coincide with our perception of prompt and after sounds. The fact that these lines are so straight tells us that our linear model is very accurate. In linear systems, the existence of two straight lines also proves that they originate from two distinct mechanisms (in this case, different types of vibration). Because the string vibrations are not sufficiently violent to materially distort the piano, the transfer rate of vertical vibrational energy to the horizontal vibrations is a constant. This explains why the ratio of prompt sound to after sound is independent of loudness; i.e., you cannot change the timbre by just playing softly. However, there is one caveat. Timbre is controlled by at least two factors: the prompt/after-sound ratio just discussed, and the harmonic content. The harmonic content does depend on loudness. When the hammer strikes a string with higher force, the string becomes more distorted, which creates more high frequency components in the sound. This higher harmonic content makes the sound brighter or harsher. In practice, the condition of the hammer controls the harmonic content much more than the loudness. Therefore, proper voicing is necessary in order to produce the pleasant piano tone, especially for loud sounds.
The unstruck string plays an important role in producing the una corda sound. This string acts as a reservoir into which the other two strings can dump their energy. Since the vibration of the 3rd string is in anti-phase (a driven string is in anti-phase with the driver), it takes the edge off the initial prompt sound and at the same time, excites vibrational modes that are different from those that result when all three are struck in unison. This is why the soft pedal in uprights doesn't work as well -- all the strings are struck even when the soft pedal is depressed.
Can you use a half soft pedal on a grand? This should not be controversial, but is. If you use a partial pedal, you will of course get a new sound. There is no reason why a pianist shouldn't be allowed to do that, and if it produces an interesting new effect, there is nothing wrong in that. However, this mode of play was not intentionally designed into the piano and I know of no composer who composed for half soft pedal on a grand. Note that extensive use of partial soft pedals on the grand will cause the string to shave off one side of the hammer. Also, it is impossible for the piano technician to regulate the piano in such a way that the third string will always miss the hammer at the same pedal travel for all the hammers at the same time. Thus the effect will be uneven, and different from piano to piano. Therefore, unless you have experimented and are trying to produce some strange new effect, half-pedaling is not recommended for the soft pedal on a grand. Nonetheless, anecdotal accounts seem to indicate that use of partial soft pedal on a grand does occur, probably because of ignorance on the part of the pianist about how it works.
In the double and single string sections, the strings have much larger diameters, so when the action moves sideways, the strings hit the side walls of the grooves, thus giving them a horizontal motion and increasing the after sound component. This mechanism is indeed fiendishly ingenious!
The need to excite large vertical normal modes for loud sounds explains why the loudest piano sounds are produced by rapid double strikes. This is why so many pieces of music with loud endings frequently finish with full, double strike, chords. Since the hammer hits the strings close to one end, the initial hit creates running waves traveling down the string. If the hammer is struck again immediately after the first strike, a new wave of energy is supplied, producing a louder sound. This second wave does not dissipate rapidly like the first wave because all available oscillation modes have already been excited. Thus the second strike produces the loudest sound that a piano can make. A third strike becomes unpredictable because the strings are now moving and the strings and hammer can be out of phase, in which case the third strike can deaden the sound.
In summary, the name soft pedal is a misnomer for a grand. Its main effect is to change the timbre of the sound. If you play a loud sound with the soft pedal depressed, it will be almost as loud as without the soft pedal. This is because you have put roughly the same amount of energy into making the sound. On the other hand, it is easier to play softly using the soft pedal on most pianos. Provided that the hammer is in good condition, you should be able to play just as softly without the soft pedal. A partial soft pedal will produce all sorts of unpredictable, uneven effects and should not be used for a grand.